【笔记】Option Greeks

基于格拉ECON5009课件的笔记；

Without dividend

红线是在到期日之前的 Value，蓝线是到期日的 Intrinsic Value，差距是 Time Value

We have:

$$\begin{align} c &= S_tN(d_1)- e^{-r (T-t)}KN(d_2)\\ p &= e^{-r (T-t)}KN(-d_2)- S_tN(-d_1) \end{align}$$

with:

$$\begin{align} d_1&=\frac{1}{\sigma \sqrt{T-t}}(ln\frac{S_t}{K}+(r+\frac{1}{2}\sigma^2)(T-t))\\ d_2&=\frac{1}{\sigma \sqrt{T-t}}(ln\frac{S_t}{K}+(r-\frac{1}{2}\sigma^2)(T-t))\\ &=d1 - \sigma \sqrt{T-t} \end{align}$$

and:

$$N'(d_1)=\frac{1}{\sqrt{2\pi}}e^{-\frac{d_1^2}{2}}\ ,\ N'(d_2)=\frac{1}{\sqrt{2\pi}}e^{-\frac{d_1^2}{2}}\frac{S_t}{K}e^{r(T-t)}$$

Delta

• Delta of a call is always positive(+); Delta of a put is always negative(-). $$\begin{align} &\Delta = \frac{\partial \Pi}{\partial S}\\ &\Delta_C = N(d_1)，\Delta_P = N(d_1)-1\\ &\Delta_C \to 1，deep\ in\ the\ money\ (S_t \to +\infty)\\ &\Delta_P \to -1，deep\ in\ the\ money\ (S_t \to 0) \end{align}$$

Gamma

$$\begin{align} &\Gamma = \frac{\partial^2 \Pi}{\partial S^2}\\ &\Gamma_C = \Gamma_P = \frac{N'(d_1)}{S_t\sigma\sqrt{T-t}} \end{align}$$

Theta (When is measured in days, divided by 252)

$$\begin{align} &\Theta = \frac{\partial \Pi}{\partial (T-t)}\\ &\Theta_C = -rKe^{-r(T-t)}N(d_2)-S_tN'(d_1)\frac{\sigma}{2\sqrt{T-t}}\\ &\Theta_P = rKe^{-r(T-t)}N(-d_2)-S_tN'(d_1)\frac{\sigma}{2\sqrt{T-t}}\\ &\Theta_C \to (0, large\ and\ negative,-rKe^{-r(T-t)})\\ &\Theta_P \to (rKe^{-r(T-t)}, large\ and\ negative,0) \end{align}$$

With BSM PDE

$$\Theta + rS\Delta + \frac{1}{2}\sigma^2S^2\Gamma = r\Pi$$

Vega

$$\begin{align} &Vega = \frac{\partial \Pi}{\partial \sigma}\\ &Vega = S_t\sqrt{T-t}N'(d_1)\\ &Vega \to (0,+,0) \end{align}$$

Rho

$$\begin{align} &Rho = \frac{\partial \Pi}{\partial r}\\ &Rho_C = K(T-t)e^{-r(T-t)}N(d_2)\\ &Rho_P = -K(T-t)e^{-r(T-t)}N(-d_2)\\ &Rho_C \to (0,+),Rho_P \to (-,0) \end{align}$$